On Level Clustering in Regular Spectra

数理解析研究所講究録 1156 巻 2000 年 113-130 113

Nariyuki MINAMI* $l’\phi’/|\tau\dot{L}^{\prime \mathrm{f}^{\nearrow}\prime}’\dot{\eta}’$ ) lnstitute of Mathematics, University of Tsukuba Tsukuba-shi, lbaraki, 305-8571 Japan

1 Introduction.

Let $H(\hslash)$ be the quantum Hamiltonian describing a bounded, isolated physical system of finite degrees of freedom. ( $\hslash$ is the Planck’s constant.) Then the spectrum of $H(\hslash)$ is purely discrete, consisting of energy levels $\{E_{n}(\hslash)\}_{n\geq 1}$ . Suppose further that $H(\hslash)$ , or its spectrum itself, is obtained by quantizing (in a certain way) the classical Hamiltonian $H(p, q)$ defining the system. Now the classical dynamical system corresponding to $H(p, q)$ may belong to one of two extreme cases of being completely integrable or chaotic. It was Percival [12] who proposed to distinguish the spectrum of $H(\hslash)$ according to the category to which the classical counterpart of $H(\hslash)$ belongs. He called the spectrum regular [resp. irregular] if $H(p, q)$ is integrable [resp. chaotic], and discussed possible difference between these two kinds of spectra. Later, Berry and Tabor [2] argued that regular and irregular spectra are distinguished by looking at the probability distribution of the spacing between adjacent energy levels. Namely they argued that in the regular spectra, the level spacing obeys the exponential distribution $e^{-t}dt$ , so that the spectrum looks like a typical realization of the Poisson point process, which is the phenomenon they called “level clustering ”. On the other hand, they conjectured that in the irregular

spectra, the level spacing distribution $p(t)dt$ satisﬁes $p(t)\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}.t^{\gamma},$ $t\searrow \mathrm{O}$ , with $\gamma>0$ (“level repulsion ”). Hence irregular spectra should typically look like the spectra of large random matrices. This conjecture is supported by numerical studies performed later. (See e.g. [3] for a review.) However, neither the precise meaning of the “probability distribution ”of level spacing, nor a mathematical formulation of the statistics for the spectrum of $H(h)$ -the level statistics-in general seems to have been explicitly given in physics litterature, although

$\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{p}_{0}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ ideas are stated in [2]. The present paper, which is an elaboration of a part of the author’s previous note [10], aims at giving a mathematical formulation of level statistics based on the idea of Berry and Tabor, and applying it to regular spectra. In \S 2, we give the definition of strict (Definition 1) and wide (Definition 2) sense level clustering, and prove solne preliminary results for later references. These results are formulated in analogy with corresponding propositions in the theory of stationary point processes. In \S 3, we shall apply the level statistics thus formulated to regular spectra, and discuss $\mathrm{t}1_{1}\mathrm{e}$ closely related theorems by Sinai [15] and Major [7]. We argue that although it is probably very difficult to apply theorems of Sinai and Major to prove the strict sense

$*$ -mail. [email protected] 114

level clustering in gelleric regular spectra, there is solne hope in proving the wide sense level clustering for solne concrete Hamiltonian such as rectangular billiard. Finally in \S 4, we shall propose another fornmlation of level statistics, and with an exalllple, shall argue again that “level clustering ”should not always mean strict Poissonian property of the

$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{C}\mathrm{t}_{\Gamma}\mathrm{u}\mathrm{l}\mathrm{n}$ .

2 Strict and wide sense level clustering. 2.1 The unfolding of Berry and Tabor.

When one speaks of the “probability distribution ”of the spacing of the adjacent ellergy levels of $H(\hslash)$ , the follwing two questions immediately arise:

1. There is no stochasticity in $H(\hslash)$ , hence in $\{E_{l},(h)\}_{n\geq 1}$ too. $\mathrm{T}\mathrm{h}\mathrm{e}1^{\cdot}\mathrm{f}_{0}\mathrm{r}\mathrm{e}$ . it will $\mathrm{b}‘\lrcorner$

$\mathrm{t}1_{1}\mathrm{e}$ necessary to take semiclassical limit $h\searrow \mathrm{O}$ to have suﬃciently many levels in a ﬁxed energy intervals, and we will have to take statistics among these levels.

$\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\ln$ 2. The lnean level density is not over the entire $\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{r}\mathrm{U}}}111$ , so that the statistical

property of $E_{n+1}-E_{n}$ may not be uniform in $n$ . Hence we will need to $\mathrm{u}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{c}|$

”the spectrunl so that it will look like a $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}_{\mathrm{o}\mathrm{r}}1\mathrm{I}\mathrm{l}\mathrm{l}\mathrm{y}$ distributed sequence.

$\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ lnspired by [2], we make a of the $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}_{\Gamma}\mathrm{u}111\{E_{l},(h)\}_{\iota\geq},1$ of $H(\hslash)\backslash \mathrm{v}\mathrm{h}\mathrm{i}(.11$ lneets the above two $\mathrm{r}\mathrm{e}\mathrm{q}_{\mathrm{U}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{I}1}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$ simultaneously. For this purpose, we lnake the following

$\mathrm{a}\mathrm{S}\mathrm{s}\mathrm{u}111\mathrm{P}^{\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}$ :

(A1) All levels are non-degenerate and $E_{n}(\hslash)\geq 0$ ;

(A2) $E_{n}(h)\searrow 0$ monotonically as $h\searrow 0$ ;

(A3) For each ﬁxed $E>0$ , there is a constant $l\text{ノ}(E\mathrm{I}>0$ such that

$N_{\hslash}(E)\equiv\#\{n\geq 1|E_{n}(\hslash)\leq E\}\sim\iota \text{ノ}(E)h^{-}d(h\searrow 0)$ . (1)

Let us take $E>0$ , which we shall fix throughout. By (A1) and (A2), we can define $\Gamma_{l},\cdot=h_{j}(E)$ as the unique solution of the equation $E_{j}(\hslash)=E$ . If we define

$\lambda_{j}=\iota \text{ノ}(E\mathrm{I}\hslash,\cdot(E)-d$ , (2)

$\mathrm{t}1_{1\mathrm{e}\mathrm{n}}\mathrm{f}_{\Gamma}\mathrm{o}111$ (A3), it is easily seen that

$n(L)\equiv\#\{j\geq 1|\lambda_{\dot{j}}\leq L\}\sim L$ $(Larrow\infty)$ . (.3)

Thus we have obtained a sequence $\{\lambda\}_{j}$ which has asymptotically uniforlIl distribution. We will call the $\lambda_{j}’ \mathrm{s}$ the unfolded levels, and call the above procedure $\zeta(\mathrm{t}\mathrm{h}\mathrm{e}$ unfolding of

$‘$ ( Berry and Tabor ”. (It is also called $\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ of Planck’s constant ”in [2].) 115

2.2 Statistics for asymptotically uniformly distributed sequences.

$\mathrm{t}1_{1}\mathrm{i}\mathrm{s}$ In subsection, we suppose that $\{\lambda,\}_{=0}^{\infty},\cdot$ is simply a strictly increasing sequence of real ( 11nlIlbers satisfying (3) and $\lambda_{0}=0$ , but still call $\lambda_{j}$ the $‘ j$ -th level ”. We introduce the

$\mathrm{f}\mathrm{o}1]0\backslash \mathfrak{j}\prime \mathrm{i}_{1\mathrm{l}}\mathrm{g}$ notation:

. $\mathit{1}j\backslash ^{\tau}(t)=N(t,\cdot c)=\#\{j\geq 1|\lambda_{j}\in(t, t+c]\}=,\cdot\sum_{\geq 1}1_{(t.\mathrm{f}}+c](\lambda i))$ (4)

$\pi_{k}(c;L)=\frac{1}{L}\int_{0}^{L}.1_{\{()}Nt:c=k\}dt$ ; (5)

$\mu_{k}(c;L)=\frac{1}{L}\int_{0}^{L}dt=,\cdot\sum_{\geq k}\pi_{j}(_{C},\cdot L)$ , (6)

$\backslash \iota’ 1\mathrm{l}\mathrm{e}\mathrm{l}\cdot \mathrm{e}$

$= \frac{1}{k!}j(j-1)\cdots(j-k+1)$ ; (7)

’ dlld $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\backslash ./$

. $\rho(c;L)=\frac{\#\{_{J}\geq 1|\lambda,\cdot\leq L,\lambda,+1-\lambda_{j}\leq C\}}{\#\{j\geq 1|\lambda_{j}\leq L\}}$ (8)

Here $c>0$ and $k=0,1,2,$ $\ldots$ . These notions have the following obvious probabilistic meanings. $N(t;c)$ is nothing but the number of levels in the interval $(t, t+c]$ . $\pi_{k}.(c;L)$ is the probability that this $\mathrm{i}_{11\mathrm{t}1}\mathrm{e}\cdot\backslash \cdot \mathrm{a}1$ catches exactly $k$ levels when $t$ is randomly cllosen from $(0, L]$ according to the $\iota 11\urcorner \mathrm{i}\mathrm{f}_{01111}$ distribution. and $\mu_{k}(c;L)$ is the k-th factorial moment of the random varible

$L$ $.\backslash ^{\gamma}(\cdot:c)$ . Finally. $\rho(c;L)$ is the relative frequency of those pairs of levels below which $11\mathrm{d}\backslash I\mathrm{e}$ spacing not exceeding $c$ . We also write

$\overline{\mu}_{\mathrm{A}}(c)=\lim\sup\mu_{k}(c;L)$ ; $\underline{\mu}_{k}(c)=\lim_{Larrow}\inf_{\infty}\mu k(C;L)$ ; (9) $Larrow\infty$

$\overline{\rho}(c)=\lim\sup\rho(c;L)$ ; $\underline{\rho}(c)=1\mathrm{i}\mathrm{r}\mathrm{n}\inf_{Larrow\infty}\rho(C;L)$ , (10)

$Larrow\infty$

$\mathrm{d}1\overline{\perp}\mathrm{d}$ also

$\pi_{k}(c)=Larrow\infty 1\mathrm{i}\ln\pi_{k}(c;L)$ ; $\mu_{k}(c)=\lim_{Larrow\infty}\mu_{k}(c;L)$ ; $\rho(c)=\lim_{Larrow\infty}\rho(c;L)$ (11)

$\iota \mathrm{v}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{e}\backslash \gamma \mathrm{e}\mathrm{r}$ these lilnits exist. This $\rho(c)$ will be called the limiting level spacing distribution

$\mathrm{f}_{111}1\mathrm{C}\mathrm{l}\mathrm{j}_{(})\mathrm{O}$ .

Proposition 1 $Unde7^{\cdot}(\mathit{3})$ . we haue $\mu_{1}(c)=c$ for any $c>0$ . 116

$P,.oof$. By the deﬁnition,

$\mu_{1}(c;L)=\frac{1}{L}\sum_{j}\int_{0}^{L}1_{[c}\lambda_{j}-,\lambda_{\mathrm{J}})(t)dt$ . (12)

But if $L>c>0$ , one has

$c$ , if $c\leq\lambda_{J}\cdot\leq L$ $\int_{0}^{L}.1_{[\lambda_{j}}-C.\lambda_{J})(t)dt=\{$ $0$ , if $\lambda,\cdot>L+c$

alld

$\int_{0}^{L}1_{[-}\lambda_{\mathrm{J}}C,\lambda)(t)jdt\leq c$ , if $\lambda_{j}

Hence for each ﬁxed $c>0$ , it is obvious that

$\mu_{1}(c;L)\sim c\frac{n(L)}{L}$ as $Larrow\infty$ ,

so that $\mu_{1}(c)=\lim_{Larrow\infty}\mu_{1}(_{C};L)=c_{L}\lim_{arrow\infty}\frac{n(L)}{L}=c$ ,

which was to be proved.

Now we give a deﬁnition of $\{\lambda,\cdot\}’ \mathrm{s}$ looking like Poisson point process in the following way:

Deﬁnition 1 We shall say that one has the strict sense level clustering if

$\pi_{k}(c)\equiv\lim_{Larrow\infty}\pi_{k}(c;L)=e^{-\mathrm{C}_{\frac{c^{k}}{k!}}}$ (13)

1, $old.\backslash for$ each $c>0$ and $k\geq 0$ ,

Note that if $\{\lambda_{j}\}_{l\geq 1}$ actually is the realization of Poisson point process on $[0, \infty)$ , then

$.\mathrm{b}\mathrm{y}$ the ergodic theorem, one has the strict sense level clustering with probability one. Obviously, the strict sense level clustering is equivalent to the weak convergence of the

$\mathrm{P}o\mathrm{i}_{\mathrm{S}}\mathrm{S}\mathrm{o}11$ family of probability distributions $P(c;L)\equiv\{\pi_{k}(c;L)\}^{\infty}k=0(L>0)$ , on $\mathrm{Z}_{+}$ to the

distribution $\{e^{-\mathrm{c}_{C}k}/k!\}_{k=0}^{\infty}$ as $Larrow\infty$ . As is well known in elementary probability $\mathrm{t}1_{1}\mathrm{e}\mathrm{o}\mathrm{r}_{\vee}$ . a suﬃcient condition for this weak convergence is that the factorial lllolllents $\mu_{k}(c:L)$ of all order of $P(c;L)$ converges to those corresponding to the Poisson distribution. Namely we have

Proposition 2 If

$\mu_{k}(_{C)}=\lim_{Larrow\infty}\mu_{k}(C,\cdot L)=\frac{c^{k}}{k!}$ (14)

$th\epsilon$ $\backslash ^{\neg}$ $Cl_{uste}ring$ hold. for each $c>0$ and $k=1,2,$ $\ldots\cdot$ , then one has strict sense level . 117

If one has the strict sense level clustering, then the limiting level spacing distribution exists and is the exponential distribution $e^{-C}dc$ . This is a direct consequence of the following lnore general proposition:

Proposition 3 If $\pi_{0}(c)=\lim_{Larrow\infty^{\pi}0}(c;L)$ exists and is diﬀerentiable with respect to $c$ , thcn $\rho(c)=\lim_{Larrow\infty^{\rho}}(C;L)$ also exists, and is given by

$\rho(c)=1+\frac{d}{dc}\pi_{0}(C)$ . (15)

In fact, if the strict sense level clustering holds, then one has in particular $\pi_{\mathrm{U}}(c)=\epsilon^{-\dot{\llcorner}}$ Hence by the above proposition, $\rho(c)=1-\epsilon^{-c}$ In \S 1 of [14], Sinai gave two definitions of $\{\lambda_{i}\}’ \mathrm{s}$ similarity to the Poisson point process. One is the strict sense level clustering as defined above, and the other is the existence of tlle limiting level spacing distribution $\rho(c)$ and its equality to the exponential distribution. The above proposition shows that these two definitions are not independent.

$\delta\in(0, c)$ $ProO.f_{\mathit{0}}fP7^{\cdot}op_{\mathit{0}}.-\backslash i\backslash tion\mathit{3}$ . If $\lambda,\cdot\leq L$ and $\lambda_{+1},\cdot-\lambda\dot,>c$ , then for any , we have the

$\mathrm{i}_{\mathrm{J}11}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$

$t\in[\lambda,\cdot-\delta,$ $\lambda\dot,)\Rightarrow N(t, \delta)>0,$ $N(t+\delta_{)}c-\delta)=0$ ,

$\backslash 1^{l}1_{1}\mathrm{e}1^{\cdot}\mathrm{e}$ lhe intervals $[\lambda_{i}-\delta, \lambda_{j})$ are disjoint. Hence if we set

$n(c;L)=\#\{j\geq 1|\lambda_{j}\leq L, \lambda_{i+1}-\lambda j>c\}\}$ then we have

$\delta n(c;L)$ $=$ $\sum_{c+1-J\lambda g>}\int-\cdot\delta L\perp[\lambda_{j}-\delta,\lambda_{\mathrm{J}})(t)dt$

$\lambda$ $\lambda_{J}\leq L$ ;

$\leq$ $\delta+\int_{0}^{L}1_{\{(\mathrm{C}}Nt;)>0,N(t+\delta;c-\delta)=0\}(t)dt$

$=$ $\delta+\int_{0}^{L}1_{\{N(\delta)\}}t+\delta;C-=0(t)dt-\int_{0}^{L}1_{\{}N(t_{\mathrm{C}}.)=\mathrm{u}\}(t)dt$ , ll.d lllelvv $\delta\frac{n(c\cdot L)}{L},\leq\frac{\delta}{L}+\pi_{0}(c-\delta;L)-\pi_{0}(C;L)$ .

$\mathrm{L}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}_{1\mathrm{l}}\mathrm{g}Larrow\infty$ and noting $n(L)\sim L$ , we get

$\delta\lim_{Larrow}\sup_{\infty}\frac{n(c,L)}{n(L)}.=\delta(1-\underline{\rho}(c\mathrm{I}\mathrm{I}\leq\pi_{0}(C-\delta)-\pi_{0(}C)$ ,

$()1^{\cdot}$

$\underline{\rho}(c)\geq 1+\frac{\pi_{0}(c)-\pi_{0}(c-\delta)}{\delta}$

Letting $\delta\searrow 0$ , we arrive at

$\underline{\rho}(c)\geq 1+\frac{d}{dc}\pi_{0}(C)$ . 118

$()\mathrm{n}$ the other hand, if $N(t, \delta)>0$ and $\mathit{4}\mathrm{V}(t+\delta;c)=0$ , then there is a $\lambda,$ $\in(t, t+\delta]$

$‘\backslash \mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}1_{1}\mathrm{a}\mathrm{t}\lambda_{+1},-\lambda,$ $>c$ . Hence the set

$\{t\in[0, L]|N(t\backslash \delta)>0 , A’\backslash ^{r}(t+\overline{\delta};c)=0\}$

is tlle $\mathrm{U}11\mathrm{i}_{0}\mathrm{n}$ of ﬁnitely lllany disjoint intervals $Ip$ each with lengtll no greater than $\delta$ , and

$\mathrm{t}1\overline{1}\mathrm{e}$ $\cdot$ $\mathrm{f}_{01}\cdot \mathrm{e}\mathrm{d}(1_{1}l_{l}$ , there corresponds a $\lambda_{j}\leq L+\overline{\delta}$ satisfying $\lambda_{\mathit{4}+1}-\lambda_{j}>c$ . Hence nulllbel of tllese intervals does not exceed $1+n(c;L)$ , and so

$\delta(\rfloor+n(c;L))$ $\geq$ $\int_{0}^{L}1_{\{N}t;\delta)>\mathrm{U}.N1^{t}+\mathit{5}:c)=\mathrm{u}1((t)dt$

$=$ $\int_{\mathrm{U}}^{L}1_{\{N}(t+\delta:C)=0\}(t)dt-./\mathrm{U}L1_{\{N(t_{C}}+^{s)}=0\}(t)\mathrm{c}lt$ .

$\lfloor)\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{i}\mathrm{l}$ by $L$ and letting $Larrow\infty$ , we have

$\delta 1\mathrm{i}111\inf_{Larrow x}\frac{n(c,L)}{n(L)}.=\delta(1-\overline{\rho}(C))\geq\pi_{0}(c)-\pi_{0}(_{C}+\overline{\delta})$ ,

$()\mathrm{I}^{\cdot}$

$\overline{\rho}(c)\leq 1+\frac{\pi_{0}(c+\delta)-\pi_{\mathrm{u}}(C)}{\delta}$

Again letting $\delta\searrow 0$ , we get

$\overline{\rho}(c)\leq 1+\frac{d}{dc}\pi_{0}(C)$ , completing the proof.

As will be explained in \S 3, it seellls to be a diﬃcult problelll to prove $\mathrm{t}1_{1}\mathrm{e}$ strict sense level clustering for any concrete $\mathrm{H}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{t}_{\mathrm{o}\mathrm{n}}\mathrm{i}\mathrm{a}\mathrm{n}$. Ill fact, no explicit exalllple of regular

$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}_{\mathrm{l}\mathrm{u}}111$ is kllown which shows strict sense level clustering. Moreover, as was shown in [8] alld xv ill be discussed in \S 4, there is an $\mathrm{e}\mathrm{X}\mathrm{a}\ln_{\mathrm{P}^{\mathrm{l}\mathrm{e}}}$ of one-dinlensiollal Hallliltonian for whicll

$\mathrm{t}\mathrm{l}\iota \mathrm{e}$ level statistics, ill a somewhat diﬀerent formulation, can be rigorously $\mathrm{p}\mathrm{e}\mathrm{l}\cdot \mathrm{f}\mathrm{o}\Gamma \mathrm{n}\mathrm{l}\mathrm{e}\mathrm{d}$ , but the obtained level spacing distribution is diﬀerent from the exponential distribution. In that case, we have still $\rho’(0+)>0$ , so that we should say that the level clustering is taking place, and if we take the high disorder limmit in the $\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{n}$ , then the data $/J(c)\mathrm{c}\mathrm{o}\mathrm{n}\vee \mathrm{e}\mathrm{l}\cdot \mathrm{g}\mathrm{e}\mathrm{S}$ to the distribution function of $e^{-C}dc$ . This situation suggests us that $\mathrm{t}1_{1}\mathrm{e}$ strict sense level

$\mathrm{c}\mathrm{l}\mathrm{t}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{I}^{\cdot}\mathrm{i}_{1}$ can only be proved in some ideal lilllit, and generically level $\mathrm{c}1_{\mathrm{u}\mathrm{S}\mathrm{t}\mathrm{e}}\mathrm{r}\mathrm{i}_{1}1^{\circ}\mathrm{S}\mathrm{h}_{\mathrm{o}\mathrm{u}}1_{\mathrm{C}}\iota \mathrm{t}\supset$

$1\iota \mathrm{o}\mathrm{t}$ lllean the strict Poissonian character of the unfolded spectrum. Thus we are led to dehne the notion of level clustering in much $\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{l}\backslash \mathrm{e}\prime \mathrm{r}$ sense, nevertheless retaining sonle

$\mathrm{p}1_{1}\mathrm{y}\mathrm{s}\mathrm{i}_{\mathrm{C}}\mathrm{a}1$ signiﬁcance. Taking the broadest statistical sense of the words $‘(\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$ ”and

$\mathrm{r}$ $‘(\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{u}1_{\mathrm{S}\mathrm{i}_{\mathrm{o}\mathrm{n}}}$ , we now make the following deﬁnition.

$s\epsilon nsel\epsilon u\epsilon lcl_{\mathcal{U}_{-\backslash ^{\neg}}}.te?\cdot ir\iota$ Deﬁnition 2 We shall say that one has th $\epsilon$ wide )$C$ [resp. $repu/.\backslash io’\iota$]

$l.J^{\cdot}$

$\lim_{c\searrow}\inf_{0}\frac{\underline{\rho}(c)}{c}>0$ $[7^{\cdot}esp$ . $\lim_{c\searrow \mathrm{U}}\sup\frac{\overline{\rho}(c)}{c}.=0]$ (16)

$hold_{\mathrm{c}}\backslash \neg$ . 119

We call give a criteria for the wide sense repulsion and clustering in terms of $\mu_{k}(c)$ , $k\leq.3$ . Recall that we always have $\mu_{1}(c)=c$ .

$ha_{-\backslash }.-th\epsilon\iota oid\epsilon.\backslash e$ Proposition 4 $(l)If\cdot\overline{\mu}2(C)=o(c^{2})$ as $c\searrow \mathrm{O}$ , then orle ns $\epsilon \mathit{1}\epsilon \mathrm{o}e/repu\mathit{1}.\backslash io\mathrm{n}$ .

$th\epsilon’\iota \mathit{0}\uparrow l\epsilon$ $\epsilon\omega id\epsilon$ (//) $l/ \int l_{\mathit{2}}(c\cdot)=\frac{1}{\mathit{2}}c^{\mathit{2}}$ .for any $c>0$ and $\overline{\mu}_{3}(c)=o(c^{\mathit{2}})$ as $c\searrow \mathrm{O}$ has th

$\backslash t’ l..\backslash \mathrm{f}/_{t\mathrm{t}^{)}\xi}/C^{\cdot}/\mathcal{U}\cdot f_{\xi}r\cdot\cdot i_{7l}g$

$\rho,.f\cdot$. Successively applying the equality

$\mu_{k}(c;L)=,\cdot\sum_{\geq k}\pi_{j}(c;L),$ $k\geq 1$ ,

xve $\mathrm{o}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{i}_{1}\iota$

$1- \pi_{\mathrm{U}}(_{C};L)=,\cdot\sum_{\geq 1}\pi_{j}(c;L)=\sum_{j=1}^{k}(-1)J^{\cdot}-1(C;L)+,\sum_{=}^{\infty}\mu j.\sum_{ik+1=0}(-1k)^{i}\pi,\cdot(c;L)$

for all $f_{\backslash }\cdot\geq 1$ . But sinse

$\sum_{\mathrm{U}i=}^{\mathrm{A}}(-1)^{i}=(-1)^{k}$ , $j\geq k+1,$ $k\geq 0$ ,

$\backslash \mathrm{v}\mathrm{e}\mathrm{h}\mathrm{d}\mathrm{V}\mathrm{e}$ the inequality

$\pi_{0}(c;L)\leq 1-\mu_{1}(c;L)+\mu_{2}(c;L)$ –. . . $+(-1)^{k}\mu k(c;L)$ (17)

$\backslash \backslash ’ 1_{1\mathrm{e}\mathrm{n}}k$ is even and

$\pi_{0}(c;L)\geq 1-\mu_{1}(c;L)+\mu_{2}(c;L)$ –. . . $+(-1)^{k}\mu k(c;L)$ (18)

$\backslash \backslash ’ 1_{1\mathrm{e}}1\mathrm{I}k$. is odd. $()_{11}$ the other hand, if $L>0$ , $\mathit{1}\mathrm{V}=n(L)+1$ , namely $\lambda_{N}\leq L<\lambda_{N+1}$ , then

$L\pi_{\mathrm{U}}(c;L)$ $=$ $\sum_{0\iota=}^{N-}(\lambda_{l+1},-\lambda,-c)1++\{(1\lambda_{p}\backslash ^{\gamma}+1-\lambda_{1}\backslash \mathrm{r}-C)+\wedge(L-\lambda_{N})\}$ ,

$=$ $\sum_{n=0}^{N-1}(\lambda_{n+1}-\lambda n-c)+N\sum_{n=0}^{-}1\{C-(\lambda,1+1-\lambda_{n})\}_{+}$

$+\{(\lambda_{N+1}-\lambda_{N}-c)_{+}\wedge(L-\lambda_{N})\}$

$\leq$ $\lambda_{\mathit{1}\mathrm{v}-}cl\mathrm{V}+C\#\{n\leq \mathit{1}\mathrm{V}-1|\lambda_{n}+1-\lambda_{n}\leq c\}+(L-\lambda_{N}\mathrm{I}$

$=$ $L-cl\mathrm{V}+CL\rho(c;L)$ .

$L$ $Larrow\infty$ $\mathrm{D}_{\mathrm{J}\mathrm{V}}\mathrm{i}\mathrm{d}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$ both sides by and letting , we obtain fronl (18),

$1+c\underline{\rho}(C)-C$ $\geq$ $\underline{\pi}_{0}(_{C)}$

$\geq$ $1-\mu_{1}(C)+\mu 2(c)-\overline{\mu}_{3}(c)$

1 2 $=$ $1-c+C\overline{2}+o(C^{2})$ , 120

when the conditions of (ii) hold. Hence we get

$\lim_{c\searrow}\inf\frac{1}{c}\underline{\rho}(C)\geq\frac{1}{2}>0$ , namely the wide sense level clustering. Now suppose the condition of (i) hold. For each $0<\delta<1$ , one has

$L\pi_{0}(C,\cdot L)$ $=$ $\sum_{n=0}^{N-1}(\lambda_{n+1}-\lambda-C)n+\sum\{cNn=0-1-(\lambda_{n}1-\lambda_{n})+\}_{+}$

$+\{(\lambda_{N+1}-\lambda N-C)_{+}\wedge(L-\lambda_{N})\}$

$\geq$ $\lambda_{N}-cN+\lambda n+1-\sum_{c\lambda_{n}\leq^{s}}\{_{C}-(\lambda_{n}+1^{-}\lambda_{n})\}+(L-\lambda N-c)$

$\geq$ $(L-c)-CN+(1-\delta)C\rho(\delta C,\cdot L)$ .

Again dividing by $L$ and letting $Larrow\infty$ and noting (17),

$1+(1-\delta)c\overline{\rho}(\delta c)-c$ $\leq$ $\overline{\pi}_{0}(c)$

$\leq$ $1-\mu_{1}(_{C})+\overline{\mu}_{2}(c)$

$=$ $1-c+o(_{C^{2})}$ .

Hence we have

$\lim_{c\searrow}\sup\frac{1}{c}\overline{\rho}(c)=\lim_{\mathrm{c}\searrow}\sup\frac{1}{\delta c}\overline{\rho}(\delta C)=^{0}$ , nalnely the wide sense level repulsion.

At this point, let us discuss the relation of the factorial moment $\mu_{k}(c)$ to the so called k-th correlation $R_{k}(c)$ deﬁned by

$R_{k}(c)= \lim_{Larrow\infty}R_{k}(C, L)$ ; (19)

$R_{k}(c,\cdot L)=\#\{(\lambda j_{1}, \ldots, \lambda j_{k})|\lambda_{j_{1}}<\cdots<\lambda_{j_{k}}\leq L, \lambda_{j_{k}}-\lambda_{j_{1}}\leq c\}$ , (20) whenever the limit exists. In particular, $R_{2}(c)$ is called the pair correlation (see e.g. [13]).

Proposition 5 (i) If $\mu_{k}(c)$ exists for each $c>0$ and is diﬀerentiable with respect to $cF$ then $R_{k}(c)$ also exists and

$R_{k}(c)= \frac{d}{dc}\mu_{k}(_{C)}$ . (21)

(ii) If $R_{k}(c)$ exists for each $c>0$ , then $\mu_{k}(c)al_{\mathit{8}}o$ exists and

’ $\mu_{k}(c)=\int_{0}^{L}.R_{k}(C’)dC$ (22) 121

$P7^{\cdot}Oof$. Since

$=$ , $\lambda_{J1}<\cdots<\lambda\sum_{\mathrm{J}k}1[\lambda_{g}c,\lambda_{J})1^{-}1(t)\cdots 1_{[-}\lambda c,\lambda_{gk})gk(t)$ we can compute

$L\mu_{k}(c;L)$ $=$ $\lambda_{\gamma_{1}}<\cdots<\sum_{gk}|[0.L]\cap \mathrm{n}_{p}[k\lambda_{\dot{J}p}-c, \lambda j_{p})|\lambda=1$

$=$

$\lambda_{J1}<\cdot.\sum_{<\lambda_{\mathrm{J}}k}.$ $\{(L \mathrm{A} \lambda_{l1})-(\lambda_{j_{k}}-c)_{+}\}_{+}$

$=$ $(c-\lambda jk^{+}\lambda_{1},\cdot)_{+}$

$\lambda_{g_{1}}\lambda_{\gamma}\sum_{<\cdot\cdot

$\lambda_{J1}\leq L,\lambda g_{k}\geq \mathrm{C}$

$+ \sum_{L<\lambda_{y1}<\cdot\cdot<\lambda Jk}.(C-\lambda jk+L\mathrm{I}++\sum_{<\lambda_{\gamma_{1}}<\cdots<\lambda_{Jk}C}\lambda_{l_{1}}$

$=$ $\lambda_{J1}<\cdots<\lambda_{J}\sum_{k}(_{C}-\lambda_{jk}+\lambda\leq Lj1\mathrm{I}+$

$+[_{\lambda_{J}<}1< \sum_{\lambda_{J1}\leq L<\lambda_{Jk}}\lambda_{J}k(c-\lambda,\cdot+\lambda_{i1})_{+}k+L<\lambda_{j}1\sum_{\lambda<\cdot\cdot

. $+< \ldots<<\sum_{\lambda_{\gamma_{1}}\lambda_{\gamma_{k}}c}[\lambda_{j}-1(c-\lambda_{ik}+\lambda j1)_{+}]$

The third $\mathrm{t}\mathrm{e}\mathrm{r}\ln$ is independent of $L$ , hence is $\mathcal{O}(1)$ as $Larrow\infty$ . On the other hand, the second ternl, which we denote by $M(\dot{L})$ , can be estimated in two ways. To begin with, we have

$\underline{<}$ $(c-\lambda_{j}\cdot k^{+}\lambda_{j_{1}})_{+}$ $M(L.\cdot)$ $1 \mathrm{J}<\cdot\cdot<\sum_{\lambda_{j}\lambda}k$

$\lambda_{J1}\leq L<\lambda_{jh}\mathit{0}\Gamma L<\lambda_{J1}<<\lambda_{J}\leq L+\mathrm{c}k$

$=$ $\int_{-\infty}^{\infty}.1_{(c]}t.t+(\lambda j1)$ . . . 1 $(t,t+\mathrm{c}](\lambda_{j})kdt$ . $\lambda_{J1}<\cdots<\lambda\sum_{J_{k}}$ ,

$\lambda_{)1}\leq L<\lambda g_{k}$ or $L<\lambda_{g_{1}}<\cdot\cdot<\lambda\leq jkL+c$

Suppose $\lambda_{1},\cdot\leq L<\lambda_{j_{k}}$ Then we have $1_{(t.t+C]}(\lambda_{j1})=0$ for $t>L+c$ and 1 $(t.t+c](\lambda_{l_{k}})=0$ $\mathrm{f}\mathrm{o}1^{\cdot}tL+c$ and 1 $(t.t+\mathrm{c}](\lambda,1)=0\mathrm{f}\mathrm{o}1^{\cdot}t

$\leq$ $M(L)$ $\sum_{\lambda_{J1}<\cdots<\lambda \mathrm{J}k}\int L^{\cdot}-c)1_{(c]}t,t+(\lambda j1\ldots 1_{(}t,t+c](\lambda j_{k})dtL+C$

$=$ $\int_{L-C}^{L+}.C(l\mathrm{V}(t)k)dt=o(L),$ $Larrow\infty$ 122

$\backslash \mathrm{v}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}\mu \mathrm{A}(c)$ exists. On the other hand, we also have the inequality

$M(L)$ $\leq$

$C \sum_{\leq\lambda_{J1}\leq L<\lambda orL<\lambda<<\lambda L+c\mathit{3}kJ1jk}1_{\{-\lambda_{g}}\lambda_{J1}<\cdots<\lambda Jk|..\lambda gk1\leq C\}$

$\leq$ . $c \sum_{\mathrm{c}\lambda_{g_{1}}<\cdots<\lambda\leq jk+L}1\{\lambda-\lambda\leq gkJ1\}c-c\lambda_{g1}\cdots<\lambda\leq J_{k}L\sum_{<}1\{\lambda_{I_{k}}-\lambda,1\leq \mathrm{C}\}$

Obviously, the right hand side is of $o(L)$ as $Larrow\infty$ when $R_{k}(c)$ exists. Therefore, under either of the conditions of (i) or (ii), one has . $L \mu_{k}(c;L)=\ldots\sum_{L\lambda_{I}<,1<\lambda gk\leq}(c-\lambda_{j_{k}}+\lambda j_{1})++o(L)$

Let us prove (i). From the above remark and

$(c+\delta-\lambda_{j}k^{+\lambda}’.1)+-(_{C}-\lambda_{l_{k}}+\lambda_{1},\cdot)+\geq\delta 1_{\{\lambda_{jk^{-\lambda_{g_{1}}\leq\}}}}c$ ’ we see

$\mu_{k}(C+\delta)-\mu k(c)\geq\delta\lim_{Larrow\infty}\sup\frac{1}{L}\sum_{k}\lambda_{j1}<\cdots<\lambda\leq \mathrm{J}L1_{\{}\lambda_{k},-\lambda\leq j1\}c$ .

Similarly, we have

$\mu_{k}(c)-\mu_{k}(_{C}-\delta)\leq\delta\lim_{Larrow}\mathrm{i}1\infty 1\mathrm{f}$ $\frac{1}{L}\sum_{\leq\lambda_{J1}<\cdots<\lambda \mathrm{J}kL}1\{\lambda_{gk^{-}}\lambda\leq J_{1}\}c$ .

$\gamma$ I )nder the $\mathrm{a}\mathrm{S}\mathrm{S}\mathrm{U}\mathrm{n}\mathrm{l}\mathrm{P}^{\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}}$ of the diﬀerentiability of $\mu_{k}(c)$ , we can let $\delta\searrow 0$ , to obtain

$R_{k}(c)= \lim_{Larrow\infty}\frac{1}{L}\lambda<\cdots<\lambda\leq J1JL\sum_{k}1_{\{\lambda_{J}k}-\lambda_{j1}\leq C\}=\frac{d}{dc}\mu_{k}(c)$ .

We turn to the proof of (ii). Let $\ell\in \mathrm{N}$ and write $\delta=c/\ell$ and $\triangle=\lambda_{k},\cdot-\lambda_{j_{1}}$ Then

$\delta\sum_{=7?\mathrm{t}1}^{l}1_{\{c}\geq\triangle+m\delta\}\leq(c-\triangle)_{+}\leq\delta\sum_{=n\iota 0}^{\ell}1\{C\geq\triangle+7n\delta\}$ .

Hence

$\leq$ $\frac{c}{\ell}\sum_{\prime n=1}^{l}\frac{1}{L},\sum\lambda_{1}<\cdots<\lambda\leq jkL1_{\{\leq C-}\lambda_{j}-k\lambda_{g}1mc/p\}$ $\frac{1}{L}\sum_{\lambda\lambda_{J1}<\cdots

$\leq$ $\frac{c}{\ell}\sum_{0m=}^{p}\frac{1}{L}\sum_{\lambda_{J1}<\cdots<\lambda_{k}\leq L},1_{\{}\lambda-\lambda_{J}\leq gk1-Cmc/\ell\}$ .

Letting $Larrow\infty$ , we obtain

$\frac{c}{\ell}\sum_{m=1}^{\ell}R_{k}(c(1-\frac{m}{\ell}))\leq\lim_{Larrow}\inf_{\infty}\mu k(c;L)\leq\lim_{Larrow}\sup_{\infty}\mu_{k}(c, L)\leq\frac{c}{\ell}\sum_{m=0}^{l}Rk(C(1-\frac{m}{\ell}))$ ,

for $\mathrm{a}\mathrm{l}1\parallel\in \mathrm{N}$ . Hence letting $\ellarrow\infty$ , we arrive at

$\mu_{k}..(c)=Larrow\infty 1\mathrm{i}\mathrm{n}1\mu_{k}(c;L)=c\int_{0}^{1}R_{k}(c(1-X))d_{X}=\int_{0}^{c}R_{k}(c’)d_{C’}$ , colllpleting the proof. 123

2.3 $\mathrm{C}_{\mathrm{o}\mathrm{n}1}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{n}$ with the theory of stationary point processes.

$\mathrm{R}$ Let us consider a point process $N_{\omega}(dx)$ on deﬁned for $\omega$ in a probability space $(\Omega, \mathcal{F}, P)$ $N$ equipped with an ergodic measure $\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{V}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{f}\mathrm{l}0-1\mathrm{w}\theta=\{\theta_{t}\}_{t\in \mathrm{R}}$ . We assume that is

$\theta$ -stationary in the sense that $N_{\theta_{t}\omega}=N_{\omega}\mathrm{o}\mathcal{T}_{t}$ , where $\tau_{t}x=x-t$ is the translation on

$\mathrm{R}r$ and that $l\mathrm{V}_{\omega}$ has no multiple points so that one can write $N_{\omega}= \sum_{j}\delta_{\lambda_{f(\omega)}}$ with

. . . $<\lambda_{-1}(\omega)<\lambda_{0}(\omega)\leq 0<\lambda_{1}(\omega)<\lambda_{2}(\omega)<\cdots$ (23)

Let us furhter suppose that $m\equiv E[N(0,1]]=1$ and that $E[(N(0,1])^{k}]<\infty$ for all $k\geq 2$ Now it is known that there is a measure $\hat{P}(d\omega)$ on $(\Omega, \mathcal{F})$ , which is called the Palm llleasure of the stationary point process $N$ , such that for all jointly measurable function . $f\cdot(\omega, .\sigma)\geq 0$ on $\Omega \mathrm{x}\mathrm{R}$ , the following formula holds: . .

$\int_{\Omega}$ $P(d \omega \mathrm{I}\int_{\mathrm{R}}$ $N_{\omega}(d_{S})f( \theta\omega, sS)=\int_{\Omega}.\hat{P}(d\omega)\int_{\mathrm{R}}dsf(\omega, S)$ . (24)

$\hat{P}$ is concentrated on the set $\hat{\Omega}=\{\omega|N(\{v\{\mathrm{o}\})>0\}$ and turns out to be a probability

$\mathrm{A}$ lneasure when $m=1$ . In this case, $P\{d\omega$ ) has an intuitive meaning of the conditional probability $P(d\omega|N(\{0\})>0)$ . (See [11] for detail.) Now we can apply the individual ergodic theorem and $(.24)$ , to prove that the following

$1^{\cdot}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$ hold with probability one:

(0)

$\lim_{Larrow\infty}\frac{1}{L}\#\{j\geq 1|\lambda_{j}(\omega)\leq L\}=\lim_{Larrow\infty}\frac{1}{L}N_{\mathrm{t}v}(0, L]=E[N_{\omega}(\mathrm{O}, 1]]=1$ ;

(i)

$\pi_{k}(c)\equiv\lim_{Larrow\infty}\frac{1}{L}\int_{0}^{L}1_{\{}N_{\omega}(t,t+c]=k\}dt=\lim_{Larrow\infty}\frac{1}{L}\int_{0}^{L}1\{N_{\theta}(0,c]=k\}dttd=P(N(0, C]=k)$ ;

(ii)

$\rho(c)\equiv\lim_{Larrow\infty}\frac{1}{L}\#\{j\geq 1|\lambda_{j}(\omega)\leq L, \lambda_{j+1}(\omega)-\lambda_{j}(\omega)\leq c\}=\hat{P}(N(\mathrm{o}, C]>0)$ ;

(iii)

$\mu_{k}..(c)\equiv\lim_{Larrow\infty}\frac{1}{L}\int_{0}^{L}.(N_{\omega}(t,\cdot t+C]k)dt=E[(N(\mathrm{o},\cdot c]k)]$ ;

(iv)

$R_{k}(c) \equiv\lim_{Larrow\infty}\frac{1}{L}\int^{L}0^{\cdot}\leq\sum_{)<\lambda(\omega\leq gkL}1\{\lambda_{Jk}(\omega)-\lambda_{j1}(\omega)C\}\lambda_{j}1(\mathrm{t}_{U})<\cdots=\hat{E}[(N(\mathrm{o}k-,\cdot 1c])]$ ,

$\hat{P}$ $\hat{E}[\cdot]$ $P$ where $E[\cdot]$ and denote the integration with respect to the measures and respec- tively. If we let

$f(\omega, s)=1_{(-\infty,0](_{S)}}1_{\{}N\omega(0,x-S]=j\}$ 124

in (24), we obtain the so called Palm-Khinchin formula:

$P(N(0, X] \leq j)=\int_{x}^{\infty}\hat{P}(N(0, s]=j)d_{S}$ . (25)

(See [6] for another approach.) This can be used to relate the right hand sides of (i) and (ii), (iii) and (iv). Indeed, letting ) $=0$ in Palm-Khinchin formula above, one obtains

$P(N(0, C]=0)= \int_{c}^{\infty}\hat{P}(N(\mathrm{o}, s]=^{\mathrm{o})}dS$ , nalllely

$\pi_{0}(_{C})=\int_{\mathrm{c}}^{\infty}(1-\rho(S))d_{S}$ ,

$()1^{\cdot}$

$\rho(c)=1+\frac{d}{dc}\pi_{0}(_{C})$ , whenever $\rho(\cdot)$ is continuous at $c$ . On the other hand,

$E[(l\mathrm{V}(0,\cdot c]k)]$ $=$ $\sum_{n\geq k}P(i\mathrm{V}(\mathrm{o}, C]\geq n)$

$=$ $\sum_{n\geq k}\int_{0}^{c}.\hat{P}(N(\mathrm{o}, S]=n-1)dS$

$=$ $\int_{0}^{c}.\hat{E}[(N(0;c.]k-1)]ds$ , namely

$\mu_{k}(C)=\int_{0}^{C}.R_{k}(S)d_{S}$ .

(Compare [5].) Hence when the sequence $\{\lambda_{j}\}$ is the typical realization of a stationary point process, then $\pi_{0}(c)$ , $\rho(c),$ $\mu_{k}(c)$ and $R_{k}(c)$ all exist with probability one, and are expressed as appropriate expectation values which are related with each other through Palm- Khinchin formula. By this observation, we are inclined to call the considerations developed in \S 2

$’(\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{S}\mathrm{t}\mathrm{i}\mathrm{c}$ point process theory ”. After all, the energy level statistics is based upon the hypothesis, often tacitly supposed, that the spectrum of a quantum Hamiltonian looks, after a suitable normalization, like a typical realization of a stationary point process. Hence the phenomenological side of the theory of energy level statistics should be developed in analogy of the theory of point processes.

3 Level statistics for regular spectra.

Suppose that the classical Hamiltonian system associated to $H(p, q)$ is completely integrable. Then one can transform the variable $(p, q)\in \mathrm{R}^{d}\cross \mathrm{R}^{d}$ into the action-angle

$=(I_{1}, \ldots , I_{d;\varphi_{1}}, \ldots, \varphi_{d})$ variable (I, $\varphi$ ) , and $H(p, q)=H(I)$ depends only on the action variable (see [1]). 125

Now following Percival ([12]) , Berry and Tabor ([2]), we shall say that $H(\hslash)$ , the quantizationof $H(p, q),\mathrm{h}\mathrm{a}\mathrm{s}$ regular spectrum if its energy levels are (approximately) given

$I_{1},$ $I_{d}$ by quantizing the action variables $\ldots$ , appearing in $H(I)$ Namely, we suppose

$\mathrm{t}$ hat the $\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}\cdot \mathrm{g}\mathrm{y}$ levels of $H(h)$ are

$E_{n}( \hslash)=H(\hslash(n+\frac{1}{4}\alpha))$ ; $n=(n_{1}, \ldots, n_{d})\in$ $\mathrm{Z}_{+}^{d}$ , (26) .

where $\mathit{0}\in \mathrm{Z}_{+}^{d}$ is the Maslov index. This procedure is called the EBK quantization after the nallles of Einstein, Brillouin and Keller, and it gives the exact $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{U}\ln$ in such

(( $1\mathrm{l}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}$ exdlllples as the rectangular billiards and the harmonic oscillators.

\ddagger $\mathfrak{s}_{11}\mathrm{d}\mathrm{e}1^{\cdot}$ lllild conditions on $H(I)$ (e.g. $H(I)$ being convex, positive with $H(\mathrm{O})=0$ ) ,

$1^{\cdot}\mathrm{e}\mathrm{g}\mathrm{t}\mathrm{l}\mathrm{d}\mathrm{l}\cdot \mathrm{b}1^{\mathrm{J}\mathrm{e}}\mathrm{C}\mathrm{t}\mathrm{r}\mathrm{a}$ satisfy the conditions $(\mathrm{A}1)^{-}(\mathrm{A}s)$ stated in \S 1 with .

$l \text{ノ}(E)=\int$ $dI1\ldots dId$ $\ldots\int_{\mathrm{R}_{+}^{d}}.1_{\{H()}I\leq E$ } , (27)

Let us apply the unfolding of Berry and Tabor as formulated in \S 2-1 to our regular

$\mathrm{b}[\mathrm{J}\mathrm{e}\mathrm{C}\mathrm{t}_{1}\cdot \mathrm{t}111$ . Note that the suﬃx $n$ distinguishing the levels is now d-dilllensional.

$1^{\mathrm{t}}\urcorner \mathrm{O}\iota\cdot x\cdot\in \mathrm{R}_{+}^{d}\backslash \{0\}$ deﬁne $h(x)=\hslash_{E}(x)\geq 0$ by the equation $H(h(X)\cdot x\cdot)=E$ , where $b^{\urcorner}>$ $()$ will be hxed throughout, and let $\lambda(x)=l\text{ノ}(E)h_{E}(X)-d$ . Our unfolded levels are $\mathrm{f}[_{\overline{1}\mathrm{e}11}$ given by $\lambda(n+\frac{1}{4}\alpha)$ , $n\in \mathrm{Z}_{+}^{d}$ Since $\lambda(\beta x)=\beta^{d}\lambda(x)$ for $\beta>0$ , we have the equivalence

$\lambda(n+\frac{1}{4}\alpha)\in(t, t+c]\Leftrightarrow??+\frac{1}{4}\alpha\in\Pi_{c}(t)$ , (28)

$\iota\backslash ’|1\in\backslash 1^{\cdot}\epsilon^{)}$ we llclve deﬁned

$\Pi_{c}(t)=\{x\in \mathrm{R}_{+}^{d}|t^{1/d}\lambda(\varphi(x))^{-1/}d<|x|\leq(t+c)^{1/d}\lambda(\varphi(x))^{-}1/d\}$ , (29)

lVitll

$\varphi(x)=\frac{x}{|x|}$

$\mathrm{v}\mathrm{o}1_{\mathrm{U}1}11\mathrm{e}$ $|\Pi_{\mathrm{c}}(t)|$ $\Pi_{\mathrm{c}}(t)$ It is $\mathrm{e}_{\mathfrak{c}}\iota \mathrm{s}_{\mathrm{J}^{\gamma}}$ to see $|\Pi_{\llcorner}(t)|=c$ , where is the of . Thus the ntllllber $N(t)=l\mathrm{V}(t;c)$ of unfolded levels in $(t, t+c]$ is equal to the nullllber

$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}$ $\frac{1}{4}\mathfrak{a}$ $\mathrm{d}_{0\Pi 1\mathrm{a}}\mathrm{i}_{1}1\Pi_{c}(t)$ of lattice $1$)( (which are shifted by ) in the , namely we have

$N(t)= \#\{1\mathrm{I}_{\mathrm{c}}(t)\cap(\mathrm{Z}_{+}^{d}+\frac{1}{4}\alpha)\}$ . (30)

As $t$ gets large, then the $\mathrm{d}_{\mathrm{o}\mathrm{n}1}\mathrm{a}\mathrm{i}\mathrm{n}\Pi_{c}(t)$ expands in the space $\mathrm{R}_{+}^{d}$ , at the sallle tilne

$\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{U}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{e}$ $\mathrm{g}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$ thillner and thinner to keep its constant. Hence provided the boundary of

$\mathrm{Z}_{+}^{d}\mathrm{w}\mathrm{o}\mathrm{t}\mathrm{l}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}1111_{\mathrm{V}}.$ $11_{\mathrm{t}}(t)$ is not too $\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}\cdot \mathrm{a}\mathrm{t}\mathrm{e}$, e.g. ﬂat, then each lattice poillt in . belong

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}_{01}11$ to $11_{\iota}(t)$ if $t\mathrm{w}\mathrm{e}\mathrm{l}\cdot \mathrm{e}$ chosen at frolll a long $\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{l}[0, L]$ , so that the total nulllber of ldttice poillts in $\Pi_{\llcorner}(t),$ nalnely $l\mathrm{V}(t)$ , would obey the Poisson distribution. This is

$\langle_{-}(J11\mathrm{c}\mathrm{e}\mathrm{i}\backslash \prime \mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$ if one recalls how Poisson’s law of slnall numbers was proved in elelllentary

$|)1([_{)}\mathrm{d}[)\mathrm{i}\mathrm{l}\mathrm{i}\dagger.\mathrm{V}$ theory. But it must be a very diﬃcult peoblem to justify this intuitive idea for concretely speciﬁed $\Pi_{c}(t)$ . In fact no example of Hamiltonian is known for which this

$\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}_{0}\mathrm{r}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{e}\mathrm{d}$ $1)\iota\cdot \mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l}$ is rigorously , to prove the strict sense level clustering. 126

At this point, it is appropriate to discuss the results of Sinai [15] and Major [7] (see

also [9] $)$ . Tlley considered the case in which $d=2$ , $\alpha=0$ and the curve (written in polar

coordillate) ? $=.f(\varphi)=\lambda(\varphi)-1/d$ which deﬁnes the boundary of $\Pi_{c}(t)$ is very random. Especially, Major proved that $\mu_{k}(c)=c^{k}/k!$ , $k\geq 1$ holds for almost all realization of the randolll curve $7^{\cdot}=f(\varphi)$ Although the randolnness thev assulne is so strong that tlle

$\mathrm{t}\cdot \mathrm{U}\mathrm{l}\cdot \mathrm{v}\mathrm{e}r=.f\cdot(\varphi)$ consisting the boundary of $\Pi_{c}(t)$ cannot be $\mathrm{S}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{O}\mathrm{o}\mathrm{t}\mathrm{h},$ $\vee \mathrm{i}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{l}$ the natural

$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{01}1$ between level statistics and the lattice points $\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{i}_{1}\overline{\perp}\mathrm{g}$ , their proof suggests us

$\mu_{k}(c)=c^{k}/k!$ $I\iota’$ $\mathrm{f}\mathrm{i}1\overline{1}\mathrm{i}\mathrm{t}\mathrm{e}I\mathrm{t}^{\Gamma}$ that it would be possible to prove for $k=1,$ $\ldots,$ with a if the

$\mathrm{d}\mathrm{i}_{111}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$ $\mathrm{b}_{\mathrm{o}\mathrm{u}\mathrm{n}}\mathrm{d}_{\mathrm{d}}\mathrm{r}.\mathrm{v}$ of $\Pi_{c}(t)$ has ﬁnite randolnness as we are going to argue now.

The $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{S}$ $\uparrow\urcorner=f(\varphi)$ , $0$ $\leq\varphi\leq\pi/2$ for the randolll curve . assunled by Sinai and Major (see [7] or [9]) are the following:

(a) $b_{1}\leq f(\varphi)\leq b_{2}$ , $|f(\varphi_{\mathit{2}})-f(\varphi_{1})|\leq b_{3}|\varphi_{2}-\varphi_{1}|$ for sonle positive constants $b,\cdot$ , $j=1,2,3$

(b) For any $k\geq 1$ , $0\leq\varphi_{1}<\cdots<\varphi_{k}$. $\leq 2\pi$ , the joint probability distribution of. $f\cdot(\varphi i)$ $,\dot{/}=1,$ $k$ $C^{1}$ $\ldots,$ has a density

$p_{k}(y_{1}, \ldots, y_{k}|\varphi_{1}, \ldots, \varphi_{k})$ .

$p_{k}(y_{1}, \ldots, y_{k}|\varphi_{1}, \ldots, \varphi_{k})\leq \mathrm{C}\mathrm{O}11\mathrm{S}\mathrm{t}.,\cdot\prod=2k(\varphi l-\varphi_{j1}-)-\mathcal{T}$

(d) Similar conditions for derivatives of $p_{k}$ .

Deﬁne

$\Pi_{c}(t\cdot, f)=\{x\in \mathrm{R}^{2}|0\leq\varphi(x)\leq\Theta , \sqrt{t}.f(\varphi(x))<|x|<\sqrt{t+c}.f(\varphi(x))\}$ . (31)

Then the area of $\Pi_{c}(t;f\cdot)$ equals $\lambda(.f)=\frac{c}{2}\int_{0}^{}f(\varphi)^{2}d\varphi$ . Let $\xi(t, f)=\#(\square _{c}(t, f)\cap \mathrm{Z}^{2})$ be the nulluber of lattice points caught in $\Pi_{c}(t, .f)$ Then the following proposition holds ([15], [7] and [9]):

Proposition 6 (i) Under the conditions (a) to (c) $above_{i}$ we haue

$\lim_{tarrow\infty}E_{P}[]=\frac{1}{k!}E_{P}[\lambda(.f)^{k}],$ $k\geq 1$ (32)

(ii) Under the conditions (a) to (d) $ab_{\mathit{0}}ve_{i}$ we have with probability one,

$\mu_{k^{\sim}}(c)=\lim_{Larrow\infty}\frac{1}{(a_{2}-a_{1})L}\int_{a_{1}L}^{a_{2}L}dt=\frac{1}{k!}\lambda(f)^{k},$ $k\geq 1$ , (33)

$\mathrm{t}p)/\iota\epsilon 7^{\urcorner}e0

We $111\mathrm{a}1_{\backslash }’\mathrm{e}$ the following observations: 127

(1) Under (b) and (c), the curve $r=f(\varphi)$ is not of $C^{2}$ , as noticed by Major [7].

(2) $\mathrm{C}^{1}o\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}1\downarrow(\mathrm{b})$ means that one has (tinﬁnite dimensional randolnness ”of the curve ’ $=.f\cdot(\varphi)$ . But the proof of the convergence of up to $k$ -th moments requires the

existence of the density $p_{k}$ only, in the assertion (i) of the above proposition, and

$p_{2k}.$. , in the assertion (ii). (3) The hardest part of the proof consists in obtaining good bounds of the number of

lattice points in $\Pi_{c}(t, f)$ for which their angles $\varphi$ are very close together. For this purpose, it is necessary to impose a condition like (c).

Now let us consider, for example, the $d$-dimensional billiard of a particle of mass 1 in the rectangle $\Lambda=\square _{j=1}^{d}[\mathrm{o}, a_{j}]$ . Then its classical Hamiltonian is $H(I)= \frac{\pi^{2}}{2}\sum_{j=1}^{d}(Ij/a_{\dot{J}})2$

, if expressed in terms of action variables, and its quantization is $H( \hslash)=-\frac{\hslash^{2}}{2}\triangle$ with the Dirichlet boundary condition on $\partial\Lambda$ . Its energy levels (eigenvalues) are exactly given by

$E_{\iota},( \hslash)=H(\hslash n)=\frac{\pi^{2}h^{2}}{2},\sum_{=1}^{d}(\frac{n}{a},’. )^{2}$ $(n=(n_{1}, \ldots.n_{d})$ , $n_{j}\geq 1)$ , (34)

dlld $\Gamma 1_{C}(t)$ is given by

$\Pi_{\mathrm{c}}(t)=\{x\in \mathrm{R}_{+}^{d}|t^{2/d}<(\frac{c_{d}}{2^{d}})(_{\dot{J}}\prod_{=1}^{d}aj)^{2}j=\sum^{/d}(\frac{x_{\dot{J}}}{a_{j}})2d1<(t+c)2/d\}$ , (35)

$\mathrm{u}$

$(a_{1}\ldots , a_{d})$ where $c_{\mathrm{r}l}$ is the volume of the unit ball. We can then conjecture that if is a $d$ -dilIlensional random variable with smooth density, then the surface of $\Pi_{c}(t)$ has d- dilllensional $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}_{0}1\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$ and for suﬃciently large. $d$ , one would have $\mu_{k}(c)=c^{k}/k!$ for $k=2$ and 3 for almost all $(a_{1}\ldots, a_{d})$ , yielding the wide sense level clustering according to Proposition 4. Before closing this section, we remark that Sarnak [13] proved $R_{2}(c)=c$ by a number tlteoretic nlethod for the quantized uniform motion on a two dimensional ﬂat torus.

$\mathrm{A}_{\mathrm{C}\mathrm{C}\mathrm{o}1}\cdot \mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ to Proposition 5 in \S 2, this is equivalent to $\mu_{2}(c)=c^{2}/2$ Unfortunately, we cannot apply Proposition 4 to conclude the wide sense level clustering in this case, because we have no information on $\overline{\mu}_{3}(c)$ .

We also relnark that if $\{\lambda_{j}\cdot\}$ is a typical realization of renewal process, namely a stationary point process 1V in which $\lambda_{n+1}-\lambda_{n}$ , $n\geq 1$ are i.i.d under the Palm measure

$\hat{P}$ , $\mathrm{t}1\overline{1}$ en $R_{\mathit{2}}(c)=c$ implies that $N$ is Poissonian. Indeed, if we set

$F(c)=\hat{P}(N(0, C]>0)=\hat{P}(\lambda_{n+1}-\lambda_{n}\leq c)$ , which does not depend on $n$ , then

$R_{\mathit{2}}(c)= \hat{E}[l\mathrm{V}(\mathrm{o}, C]]=\sum\hat{P}(N(0, c]n\infty=1\geq n)=\sum_{n=1}^{\infty}\hat{P}(\sum^{n-1}j=0(\lambda j+1-\lambda j\leq c)=\sum_{n=1}^{\infty}(F^{n*})(c),$

where $F^{n*}$ is the $n$ -fold convolution of $F$ . Hence if $\mathcal{L}(\xi)=\int_{0}^{\infty}e^{-\xi}dCF(C)$ is the Laplace transform of $dF(c)$ , then from $R_{2}(c)=c$ , we get

$\sum_{n=1}^{\infty}(L(\xi))^{n}=\frac{L(\xi)}{1-L(\xi)},=\int_{0}^{\infty}.\epsilon^{-}d\epsilon_{C}c=\frac{1}{\xi}$ 128

odlllely $\mathcal{L}(\xi)=1/(1+\xi)$ . Hence $dF(c)=e^{-C}dc$ . But the Poisson $\mathrm{p}\mathrm{o}\mathrm{i}_{1}\overline{\perp}\mathrm{t}$ process is the only renewal process for which $\lambda_{n+1}-\lambda_{n}$ obeys the exponential distribution.

4 An example of level statistics, in which non-Poissonian level clustering is observed.

(’( $1\mathrm{l}\mathrm{S}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{l}$ the $\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{d}\mathrm{i}_{1}11\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}1$ Schr\"odinger operator

$H_{v}( \hslash)=-\hslash^{2}\frac{d^{2}}{dx^{2}}+v\sum_{\dot{J}=1}^{n}\delta(x-x_{j})$ $0\leq x\leq 1$ (36)

$\backslash \iota^{r}$ ith Dirichlet boundary condition at $x=0,1$ . Here $v>0$ and

$0=x_{0}

$/ \sigma_{n}(h)=\frac{1}{\pi}\sqrt{E_{n}(\hslash)}$ Let $E_{\iota},(\hslash\backslash )$ be the energy levels of $H_{v}(h)$ and let . Then

$\#\{n|\kappa_{n}(\hslash)\leq E\}\sim\frac{E}{\hslash}$ , $E/\hslasharrow\infty$ . (37)

It is not obvious if Assumption (A2) holds, but the above asymptotics is stronger tllall

$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{S}\backslash$ $\{\kappa_{i},(\hslash)\}$ $-\mathrm{i}_{\mathrm{S}\mathrm{S}n11}1\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ (A3), so we shall directly consider the for . We can prove the following ([8]): Let $0\leq a_{1}

$l\mathrm{V}_{\hslash}(t;c)=\#\{n|\kappa_{n}(\hslash)\in(t, t+c\hslash]\}$ . (38)

Then the limit

$\pi_{k}(c)=\lim_{\hslash\searrow 0}\frac{1}{a_{2}-a_{1}}\int_{a_{1}}^{(\mathrm{J}2}1\{\mathit{1}\mathrm{V}_{\hslash}(t.c)=k\}dt$ , $k\geq 1$ (39)

$n$ always exists, and when $y,\cdot=x_{l+1}-x_{j}$ , $j=0,1,$ $\ldots,$ are rationally independent, they

$\mathrm{d}\mathrm{r}\mathrm{e}$ explicitly computable. In particular

$\pi_{\mathrm{U}}(c)=,\cdot\prod_{=0}^{n}(1-c\mathrm{L}J,\cdot)$ (40)

$?l$ $c\iota/,$ $j=0,1\ldots,$ {Or $c>0$ such that $<1,$ $,$ . Hence by Proposition 3 in \S 2.

$\rho(c)$ $=$ $1-(, \cdot\sum^{n}=0\frac{y_{\dot{j}}}{1-cy},\cdot),\prod_{=0}^{n}(1-Cy,\cdot)$

$c(1-, \cdot\sum_{0=}^{n}y^{2},\cdot),$ $c\searrow 0$

so that $\rho’(0+)=1-\sum^{n},\cdot=0y^{2},\cdot>0$ . Hence we have level clustering, but the level spacing distribution $d\rho(c)$ is diﬀerent from the exponential distribution. 129

$X_{1}.X_{2},$ $(0,1)$ Now let $\ldots$ be i.i.d random variables uniformly distributed in , and let

$- 1_{\mathrm{L}1’}^{\prime(\mathrm{L}})<\cdots

$=\lambda^{\gamma(n)},\cdot$ $x,$ , $j=1,$ . .. , $n$ , in (36). Then it can be shown that with probability one,

$, \lim_{\iotaarrow\infty}\pi_{k}(C;x_{1}(n), \ldots , X_{n}^{(/\iota)})=\epsilon^{-C}\frac{c^{k}}{k!}$ , $k\geq 0$ (41)

$\perp\backslash \mathrm{l}\mathrm{c})1^{\cdot}\mathrm{e}()\mathrm{v}\mathrm{e}\mathrm{l}\cdot$ ,

$X_{n}^{(n)})=1-\epsilon^{-c}$ $, \lim_{tarrow\infty}\rho(c;x_{1}^{(n})\ldots.,$ (42)

$\mathrm{T}1_{1}n\mathrm{s}$ we obtain Poisson distribution in the high disorder limit.

$\backslash \backslash /\mathrm{e}$ note that considering $\kappa_{\iota},(h)$ instead of $E_{\iota},(h)\mathrm{a}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{O}\mathrm{U}\mathrm{n}\mathrm{t}_{\mathrm{S}}$ to unfolding the $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{u}\ln$

$\{f,\lrcorner’,\iota(f7)\}$ so that it will have $\mathrm{a}\mathrm{s}\mathrm{y}_{111}\mathrm{p}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}_{\mathrm{o}\mathrm{r}}\mathrm{n}1$ distribution with mean density $1/\hslash$ ,

$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{l}(‘ 37)$ $\hslash\searrow 0$ $\mathrm{a}’ \mathrm{c}_{\dot{C}}111$ be seen . We then take the limit to accomplish the level statistics.

This $1^{\mathrm{j}1\mathrm{O}\mathrm{C}}.\mathrm{e}\mathrm{d}_{\mathrm{U}1}\cdot \mathrm{e},$ colIlpared to the unfolding of Berry and Tabor, looks more natural in

$\mathrm{b}\lfloor)\mathrm{i}_{1}\cdot \mathrm{i}\mathrm{t}$ , but if we apply it to regular spectra, the connection of the level statistics to lattice points counting is not as clear-cut as in \S 3.

References

$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{a}1111\mathrm{e}\mathrm{t}\mathrm{h}_{0}\mathrm{d}\mathrm{s}$ $[1].\mathrm{A}_{111()}.]\mathrm{d}$ . V.I.: of classical mechanics. Springer (1978)

[2] Berry, M.V., Tabor, M.: Level clustering in tlle regular spectrum. Proc. R. Soc. Lond.

$\vdash\backslash$ . vo1.356 (1977) 375-394

[.3] (). Bohigas, M. J. Giannoni: $\mathrm{C}^{1}\mathrm{h}\mathrm{a}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{c}$ motion and $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\ln\coprod 1\mathrm{a}\mathrm{t}_{\mathrm{l}\mathrm{i}\mathrm{x}}$ theories. in:

$.\mathrm{J}$ . S. Dehesa et al. $(\mathrm{e}\mathrm{d}\mathrm{s}.)\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}$ and colllputational methods in nuclear

[ $)[_{1}\iota\cdot \mathrm{b}\mathrm{i}\mathrm{c}\mathrm{S}$ . $\mathrm{I}_{\lrcorner}\mathrm{e}\mathrm{c}\mathrm{t}$ . Notes in Phys. vol.209, 1-99 (1984)

$[\swarrow\downarrow](^{\mathrm{t}}1_{1\langle_{\rangle}^{1}}\mathrm{n}\mathrm{g}$, Z., Lebowitz, J.L., Major, P.: On the nulnber of lattice points between two

$\mathrm{t}^{\mathrm{y}}111\mathrm{a}1_{\circ}^{0}\mathrm{e}\mathrm{d}$ and randolnly shifted copies of an oval. Probab. Th. Rel. Fields vol.100 (1994) 253-268

$\mathrm{O}_{1\overline{1}}$ $\Gamma \mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$ $[\ulcorner)](^{\mathrm{t}}\mathrm{r}\prime \mathrm{d}111\acute{\mathrm{e}}1\backslash$ , H.. Leadbetter, M.R., Serﬂing, R..J.: dist function-nloment re-

$\cdot$ $\mathrm{W}\mathrm{a}\mathrm{h}\mathrm{l}\cdot \mathrm{S}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{C}\mathrm{h}\mathrm{k}\mathrm{e}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{e}$ $1\mathrm{d}\mathrm{l}\mathrm{i}_{01}1\mathrm{b}11\mathrm{i}|\mathrm{J}\mathrm{s}$ ill a statiollal y point $\mathrm{P}^{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{S}}\mathrm{S}$ . Z. verw. Geb.

$\backslash \cdot()].\rfloor_{(\backslash }-,$ $(1^{(}J7111- 8$

[6] D..J. $\mathrm{D}_{\mathrm{d}}1\mathrm{e}\mathrm{y}$, D. Vere-Jones: All Introduction to the Theory of Point processes. Springer

$\mathrm{V}\mathrm{e}\mathrm{I}^{\cdot}\mathrm{l}\mathrm{d}\mathrm{g}$, New-York, 1988

$\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}$ $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}_{0}111$ $\cdot$ [7] Majol , P.: Poisson law for the nulllber of lattice $1$) in a strip with hnite

$\mathrm{P}\mathrm{l}\cdot \mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}$ $\subset 11^{\cdot}\mathrm{e}\mathrm{d}$ . . Th. Rel. Fields vol.92 (1992) 423-464

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}^{\mathrm{l}\mathrm{e}}1\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}$ [8] $\mathrm{M}\mathrm{i}_{11}\mathrm{d}\mathrm{I}11\mathrm{i}$ . N.: Level clustering ill a $\mathrm{f}\mathrm{i}_{11}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{n}$ . Progr. Theol. Phys. $.\mathrm{v}_{0}^{7}.1\mathrm{L}6$ (1994) 359-368

[9] Mill.d llli, N.: $011$ the Poisson limit theorems of Sinai and Major. preprint 130

[1 $()\underline{\rceil}\perp\backslash |\mathrm{l}\mathrm{i}\mathrm{D}\dot{\mathrm{e}}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{l}$. N.: Level statistics for quantum Hamiltonians- Some preliminary ideas toward mathelnatical justiﬁcation of the theory of Berry and Tabor. to appear in the

$P,.\mathrm{o}ce\epsilon di\gamma\iota g.\backslash$ in $th\epsilon s_{\epsilon Con}dC_{on}gr\epsilon s3\gamma$ ISAAC.

$[1 |\rfloor.]$ . Neveu: Processus $\mathrm{P}\mathrm{o}11\mathrm{C}\mathrm{t}_{\mathrm{U}}\mathrm{e}\mathrm{l}\mathrm{s}$. Lect. Notes ill Math. 598, 249-445. 1976

$\mathrm{L}\mathit{2}\mathit{2}9^{-}\mathrm{L}232$ [12] $\mathrm{p}_{\mathrm{e}\mathrm{l}\mathrm{C}}\mathrm{i}\iota^{r}\mathrm{a}1.1.\mathrm{C}^{\mathrm{t}}.$ : Regular and irregular spectra. J. Phys. B. vo1.6 (1973)

[13] $\mathrm{b}_{\llcorner}.\mathrm{d}1^{\cdot}11^{\cdot}\mathrm{d}1\backslash ’$. P.: Values at integers of binary quadratic forlns. Ha’monic analysis and $?lu/\gamma lbe’$ . thory CMS Conf. Proc. vo1.21, AMS (1997) 181-203

$\mathrm{i}\mathrm{n}:.7.$ [14] Ya. $(_{1}^{\tau}$ . Sinai: Mathematical problenls in the $\mathrm{t}1_{1\mathrm{e}o\mathrm{r}}\mathrm{y}$ of quantulIl chaos. Lillden-

$\mathrm{i}1\downarrow$ $\mathrm{a}11\mathrm{a}\mathrm{l}\mathrm{y}_{\mathrm{S}}\mathrm{i}\mathrm{S}$ $(\mathrm{e}\mathrm{d}\mathrm{s}.)\mathrm{c}_{\mathrm{e}\mathrm{o}\mathrm{n}1}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}$ $\mathrm{s}\mathrm{t}\mathrm{l}\cdot’\epsilon \mathrm{i}\mathrm{u}\mathrm{S}\mathrm{s}$, V. D. Millllan aspects of functional . Lect. Notes Math. vol.1469, 41-59 (1991) and in: D. K. $\mathrm{C}^{\mathrm{t}}\mathrm{d}111\mathrm{p}\mathrm{b}\mathrm{e}11$ (ed.) Chaos, Alllerican Inst. Phys. (1990)

[15] Sinai, Ya.G.: Poisson distribution in a geonletric problem. Adv. Sov. Math. (AMS) vol.3 (1991) 199-214